Distance between a Ship and a Lighthouse?

Question

The bearing of a lighthouse from a ship was found to be N 37 degrees E. After the ship sailed 2.5 miles due south, the new bearing was N 25 degrees E. Find the distance between the ship and the lighthouse at each location.

The bearing of a lighthouse from a ship was found to be N 37 degrees E. After the ship sailed 2.5 miles due south, the new bearing was N 25 degrees E. Find the distance between the ship and the lighthouse at each location.
Answer : 5.1 miles for 1st location and 7.2 for 2nd location
(show calculations)

Answer 1

If I'm understanding your terminology, N 37 deg E means 37 degrees measured CW from the north line. If so then draw a triangle with points O, L and S representing the original position of the ship, the lighthouse and the new ship position respectively.

OS = 2.5
angle L = 12
angle S = 25
andgle O = 180 - 37 = 143

We could use the sine rule to find OL and SL.
OL/sin25 = 2.5 / sin12 = SL / sin143
OL = 5.08 miles
SL = 7.24 miles

Answer 2

Start with a diagram. Put the lighthouse at one location and don't move it. The first position of the ship sees the lighthouse 37 degrees east of true north. Place the position of the ship such that 37 degrees is the measure of the angle between true north and the lighthouse. Now, place the ship due south of the first position such that the angle between true north and the lighthouse measures 25 degrees. The distance between the two positions of the ship is 2.5 miles.

The diagram you just drew is a triangle with angles 25 degrees, 143 degrees (90-37 + 90), and 12 degrees (180-25-143) with the length of one leg equal to 2.5 miles.

you can approximate the lengths of the other legs using a protractor and ruler or you can use trig to calculate the exact distance. I don't have neither so I can't give you numerical answers, just the way to solve the problem.

Answer 3

You should always draw a picture of the problem.
From my look-see, there is some mistake. For the bearing to become a lesser value, the ship would have to sail NORTH.

Answer 4

y + 2.5 = xtan37
y = xtan25
(tan37 - tan25)x = 2.5
x = 2.5/(tan37 - tan25)
d1 = 2.5/[cos37(tan37 - tan25)]
d1 = 10.9 mi.
d2 = 2.5/[cos2537(tan37 - tan25)]
d2 = 9.6 mi.
edit:
Oops!
x = ytan37 = (y + 2.5)tan25
y(tan37 - tan25) = 2.5tan25
y = 2.5tan25/(tan37 - tan25)
d1 = y/sin37 = 2.5tan25/[sin37(tan37 - tan25)]
d1 = 14.46 mi.
d2 = (y + 2.5)/sin25 = 2.5tan25/[sin25(tan37 - tan25)]
d2 = 15.52 mi.